METHODS THAT WORK TO SIMPLIFY THE STUDY OF TOTAL DIFFERENTIALS FOR FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES. SYMBOLIC VALUES OF THESE DIFFERENTIALS

Dennis M. Cates ()

Chapter Chapter 14 in Cauchy's Calcul Infinitésimal, 2019, pp 71-75 from Springer

Abstract: Abstract Let $$u=f(x, y, z, \dots )$$ always be a function of several independent variables $$ x, y, z, \dots ; $$ and, denote by $$\begin{aligned} \varphi (x, y, z, \dots ), \ \ \ \ \ \chi (x, y, z, \dots ), \ \ \ \ \ \psi (x, y, z, \dots ), \ \ \ \ \ \dots \end{aligned}$$ its first-order partial derivatives relative to x, to y, to z, $$\dots . \ $$ If we make, as in the eighth lecture, $$\begin{aligned} F(\alpha )=f(x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots ), \end{aligned}$$ then, differentiate the two members of equation (1) with respect to the variable $$\alpha , $$ we will findLinearization of multiple variable function $$\begin{aligned} \left\{ \begin{aligned} \ F^{\prime }(\alpha )&= \varphi (x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots ) dx \\&\quad +\chi (x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots ) dy \\&\quad +\psi (x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots ) dz \\&\quad + \cdots . \end{aligned} \right. \end{aligned}$$ If, in this last formula, we set $$\alpha =0, $$ we will obtain the following $$\begin{aligned} \left\{ \begin{aligned} \ F^{\prime }(0)=\varphi (x, y, z, \dots ) dx&+\chi (x, y, z, \dots ) dy \\&+\psi (x, y, z, \dots ) dz + \cdots =du, \end{aligned} \right. \end{aligned}$$ which is in accordance with equation (16) of the eighth lecture.

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_14

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